Optimal. Leaf size=71 \[ -x+2 \sqrt{3} \sqrt{2 x-3}+10 \log \left (x+\sqrt{3} \sqrt{2 x-3}+4\right )-21 \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right ) \]
[Out]
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Rubi [A] time = 0.188021, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -x+2 \sqrt{3} \sqrt{2 x-3}+10 \log \left (x+\sqrt{3} \sqrt{2 x-3}+4\right )-21 \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(12 - x)/(4 + x + Sqrt[-9 + 6*x]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ 2 \sqrt{3} \sqrt{2 x - 3} + 10 \log{\left (2 x + 2 \sqrt{3} \sqrt{2 x - 3} + 8 \right )} - \frac{21 \sqrt{6} \operatorname{atan}{\left (\sqrt{2} \left (\frac{\sqrt{2 x - 3}}{4} + \frac{\sqrt{3}}{4}\right ) \right )}}{2} - \int ^{\sqrt{2 x - 3}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((12-x)/(4+x+(-9+6*x)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0452376, size = 70, normalized size = 0.99 \[ \frac{1}{6} (9-6 x)+2 \sqrt{6 x-9}+10 \log \left (6 x+6 \sqrt{6 x-9}+24\right )-21 \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(12 - x)/(4 + x + Sqrt[-9 + 6*x]),x]
[Out]
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Maple [A] time = 0.006, size = 54, normalized size = 0.8 \[ 2\,\sqrt{-9+6\,x}+{\frac{3}{2}}-x+10\,\ln \left ( 24+6\,x+6\,\sqrt{-9+6\,x} \right ) -{\frac{21\,\sqrt{6}}{2}\arctan \left ({\frac{\sqrt{6}}{24} \left ( 6+2\,\sqrt{-9+6\,x} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((12-x)/(4+x+(-9+6*x)^(1/2)),x)
[Out]
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Maxima [A] time = 0.801669, size = 69, normalized size = 0.97 \[ -\frac{21}{2} \, \sqrt{6} \arctan \left (\frac{1}{12} \, \sqrt{6}{\left (\sqrt{6 \, x - 9} + 3\right )}\right ) - x + 2 \, \sqrt{6 \, x - 9} + 10 \, \log \left (6 \, x + 6 \, \sqrt{6 \, x - 9} + 24\right ) + \frac{3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 12)/(x + sqrt(6*x - 9) + 4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.266952, size = 90, normalized size = 1.27 \[ -\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} x + 21 \, \sqrt{3} \arctan \left (\frac{1}{12} \, \sqrt{3}{\left (\sqrt{2} \sqrt{6 \, x - 9} + 3 \, \sqrt{2}\right )}\right ) - 10 \, \sqrt{2} \log \left (x + \sqrt{6 \, x - 9} + 4\right ) - 2 \, \sqrt{2} \sqrt{6 \, x - 9}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 12)/(x + sqrt(6*x - 9) + 4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{x + \sqrt{3} \sqrt{2 x - 3} + 4}\, dx - \int \left (- \frac{12}{x + \sqrt{3} \sqrt{2 x - 3} + 4}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((12-x)/(4+x+(-9+6*x)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.269275, size = 117, normalized size = 1.65 \[ -\frac{1}{6} \, \sqrt{3} \sqrt{2}{\left (10 \, \sqrt{3} \sqrt{2}{\rm ln}\left (33\right ) - 63 \, \arctan \left (\frac{1}{4} \, \sqrt{3} \sqrt{2}\right )\right )} - \frac{21}{2} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{1}{12} \, \sqrt{3} \sqrt{2}{\left (\sqrt{6 \, x - 9} + 3\right )}\right ) - x + 2 \, \sqrt{6 \, x - 9} + 10 \,{\rm ln}\left (6 \, x + 6 \, \sqrt{6 \, x - 9} + 24\right ) + \frac{3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 12)/(x + sqrt(6*x - 9) + 4),x, algorithm="giac")
[Out]